I have to say that the answers below are great. You can't do any better than learning what Berger, Donaldson, Gromov, and Yau think are the important open problems, and each definitely has his own unique perspective on the subject.
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Deane YangJan 4 '11 at 15:17

Retagged. "Tag-removed" is only supposed to be used when there are no other tags.
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Douglas ZareFeb 10 '11 at 21:29

There are many surveys and books with open problems, but it would be nice to have a list of a dozen problems that are open and yet embarrasingly simple to state. A list that is "folklore" and that every graduate student in differential geometry should keep in his/her pocket.

Here are the ones I like best:

1.Does every Riemannian metric on the $3$-sphere have infinitely many prime closed geodesics? Does it have at least three (prime) closed geodesics?

2.If the volume of a Riemannian $3$-sphere is equal to 1, does it carry a closed geodesic whose length is less that $10^{24}$? Same question with
$S^1 \times S^2$ if you like it better than the $3$-sphere.

The outstanding problem then, in 4-manifold topology, is to find if there is something which could play the role of Thurston’s geometrization conjecture, for the case of 3-manifolds, and which might guide further research.

This is a great list, but maybe a little outdated by the other suggestions?
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Deane YangJan 4 '11 at 22:56

1

Yes, in fact Yau wrote the "Review of Geometry and Analysis" in 2000 when I was his student precisely as an updated, corrected, and massively expanded sequel to that "Problem Section".
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Spiro KarigiannisJan 4 '11 at 23:38

Surely there may be some redundancies with the other lists and I even guess that some problems from Yau's list are solved by now, but since it also contains some classic problems (like Hopf's conjecture) I thought might be worth mentioning it.
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Daniel PapeJan 4 '11 at 23:38

One of Yau’s problems is about bounded harmonic functions, and harmonic functions on noncompact manifolds of polynomial growth. After proving non-existence of bounded harmonic functions on manifolds with positive curvatures, he proposed the Dirichlet problem at infinity for bounded harmonic functions on negatively curved manifolds, and then proceeded to harmonic functions of polynomial growth. Dennis Sullivan tells a story about Yau's geometric intuition, and how it led him to reject an analytical proof of Sullivan's. Michael Anderson independently found the same result about bounded harmonic function on simply connected negatively curved manifolds using a geometric convexity construction.

Rank rigidity of nonpositively curved manifolds

Again motivated by Mostow's strong rigidity theorem, Yau called for a notion of rank for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics. Advances in this direction have been made by Ballmann, Brin and Eberlein in their work on non-positive curved manifolds, Gromov's and Eberlein's metric rigidity theorems for higher rank locally symmetric spaces and the classification of closed higher rank manifolds of non-positive curvature by Ballmann and Burns-Spatzier. This leaves rank 1 manifolds of non-positive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.

Kähler–Einstein metrics and stability of manifolds

It is known that if a complex manifold has a Kähler–Einstein metric, then its tangent bundle is stable. Yau realized early in 1980s that the existence of special metrics on Kähler manifolds is equivalent to the stability of the manifolds. Various people including Simon Donaldson have made progress to understand such a relation.

Mirror symmetry

He has collaborated with string theorists including Strominger, Vafa and Witten, and as post-doctorals from theoretical physics with B. Greene, E. Zaslow and A. Klemm . The Strominger–Yau–Zaslow program is to construct explicitly mirror manifolds. David Gieseker wrote of the seminal role of the Calabi conjecture in relating string theory with algebraic geometry, in particular for the developments of the SYZ program, mirror conjecture and Yau–Zaslow conjecture.